($(1-|\xi|^2)_+^\delta$ are known as Bochner-Riesz multipliers.) We are interested in the $L^p$ boundedness of $T_\delta$. The case $\delta = 0$ has been solved since 1971, once Fefferman provided a proof that $L^p$ boundedness fails in dimension $n \geq 2$ when $p \neq 2$. For general $\delta > 0$, a Theorem due to Herz shows that a necessary condition for boundedness is that

$
|\frac{1}{p}-\frac{1}{2}| < \frac{2\delta + 1}{2n}.
$

It's thus natural for one to conjecture that this is also a sufficient condition.

Here's what I think I know about current progress, from jotted down notes:

1 Answer
1

The results stated in your post are improved in the recent work of Bourgain and Guth, in dimensions 5 and higher. The numerology is the same as for the restriction problem for the sphere (see the statement of theorem 1 in that paper). In the case of the restriction problem for the sphere Bourgain and Guth improved the 3 dimensional result to $p > 3 + 3/10=3.3$, which is slightly better than the $p>10/3=3.333...$ from Tao's bilinear estimates (which, combined with Lee's work, gives the corresponding result for Bochner-Riesz.) This is the "same" $p>10/3$ you mention in your post. It seems likely that Bourgain and Guth's argument will give a similar improvement for the 3 dimensional Bochner-Riesz problem, but they do not work this out in the paper. They do write (page 5) "Thus in principle, one could expect the proof of Theorem 2 to carry over and lead to the validity of the Bochner-Riesz conjecture for $max(p, p′) ≥ 3 +3/10$ , if n = 3. We do not pursue the details of this matter here."